On orthonormal Muntz-Laguerre filters

Recently an upsurge in research relating to Laguerre filters for use in system optimization [1], system identification [2] and reduced-order modeling [3] has been noticed. The main reason for the good performance of Laguerre filters is that they form a uniformly bounded orthonormal basis in Hardy space [1]. Another reason, and this represents the novelty of the work carried out in this paper, is that there is a fundamental link with the well-known Müntz-Szász theorem [4]—[8]. Here we show that the ordinary Laguerre filters belong to the more general class of Müntz-Laguerre filters, which are closely related to the Müntz-Legendre quasi-polynomials [7]. We prove that when the Müntz-Szász condition [6] holds, the Müntz-Laguerre filters form a uniformly bounded orthonormal basis in Hardy space. This has consequences in terms of optimal pole-cancellation schemes and the results also imply a generalization of Lerch’s theorem [9].